Question

3. Suppose that ƒ : (0, ∞) → R is differentiable. Is each of the following claims true or false? If it is false,show it with a counterexample. If it is true, prove it. \text { (a) If } \lim _{x \rightarrow \infty} f^{\prime}(x)=1 \text {, then } \lim _{x \rightarrow \infty} f(x)=\infty \text {. } \text { If } \lim _{x \rightarrow \infty} f^{\prime}(x)=0 \text {, then } \lim _{x \rightarrow \infty} f(x)=L \text { for some } L \in \mathbf{R} \text {. } \text { If } \lim _{x \rightarrow \infty} f(x)=1 \text {, then } \lim _{x \rightarrow \infty} f^{\prime}(x)=0 \text { If } \lim _{x \rightarrow \infty} f^{\prime}(x)=1 \text {, then } \lim _{x \rightarrow \infty} \frac{f(x)}{x}=1 \text {. } \text { ) If } \lim _{x \rightarrow \infty} \frac{f(x)}{x}=1, \text { then } \lim _{x \rightarrow \infty} f^{\prime}(x)=1

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